Para hallar los extremos hay que resolver la ecuación
dxdf(x)=0(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
dxdf(x)=primera derivada∣x∣cos(x)−x2sin(x)sign(x)=0Resolvermos esta ecuaciónRaíces de esta ecuación
x1=−86.3822220347287x2=−4.49340945790906x3=−42.3879135681319x4=−32.9563890398225x5=−67.5294347771441x6=−20.3713029592876x7=23.519452498689x8=61.2447302603744x9=7.72525183693771x10=−92.6661922776228x11=14.0661939128315x12=−45.5311340139913x13=48.6741442319544x14=70.6716857116195x15=64.3871195905574x16=95.8081387868617x17=36.1006222443756x18=−29.811598790893x19=−76.9560263103312x20=−10.9041216594289x21=98.9500628243319x22=76.9560263103312x23=45.5311340139913x24=39.2444323611642x25=−98.9500628243319x26=−89.5242209304172x27=−61.2447302603744x28=17.2207552719308x29=92.6661922776228x30=4.49340945790906x31=54.9596782878889x32=−394.267341680887x33=−48.6741442319544x34=67.5294347771441x35=−7.72525183693771x36=−17.2207552719308x37=86.3822220347287x38=32.9563890398225x39=−26.6660542588127x40=26.6660542588127x41=80.0981286289451x42=108.375719651675x43=−95.8081387868617x44=20.3713029592876x45=−83.2401924707234x46=10.9041216594289x47=83.2401924707234x48=−202.627791039417x49=89.5242209304172x50=29.811598790893x51=58.1022547544956x52=−54.9596782878889x53=−64.3871195905574x54=−39.2444323611642x55=−14.0661939128315x56=−70.6716857116195x57=−73.8138806006806x58=73.8138806006806x59=−36.1006222443756x60=−58.1022547544956x61=42.3879135681319x62=−51.8169824872797x63=−23.519452498689x64=51.8169824872797x65=−80.0981286289451Signos de extremos en los puntos:
(-86.38222203472871, 0.0115756804584678)
(-4.493409457909064, 0.217233628211222)
(-42.38791356813192, 0.0235850682290164)
(-32.956389039822476, -0.0303291711863103)
(-67.52943477714412, 0.0148067339465492)
(-20.37130295928756, -0.0490296240140742)
(23.519452498689006, -0.0424796169776126)
(61.2447302603744, -0.0163257593209978)
(7.725251836937707, 0.128374553525899)
(-92.66619227762284, 0.0107907938495342)
(14.066193912831473, 0.0709134594504622)
(-45.53113401399128, -0.0219576982284824)
(48.674144231954386, -0.0205404540417537)
(70.6716857116195, 0.0141485220648664)
(64.38711959055742, 0.0155291838074613)
(95.8081387868617, 0.0104369581345658)
(36.10062224437561, -0.0276897323011492)
(-29.81159879089296, 0.0335251350213988)
(-76.95602631033118, -0.0129933369870427)
(-10.904121659428899, 0.0913252028230577)
(98.95006282433188, -0.010105591736504)
(76.95602631033118, 0.0129933369870427)
(45.53113401399128, 0.0219576982284824)
(39.24443236116419, 0.0254730530928808)
(-98.95006282433188, 0.010105591736504)
(-89.52422093041719, -0.0111694646341736)
(-61.2447302603744, 0.0163257593209978)
(17.22075527193077, -0.0579718023461539)
(92.66619227762284, -0.0107907938495342)
(4.493409457909064, -0.217233628211222)
(54.959678287888934, -0.0181921463218031)
(-394.26734168088706, 0.00253634191261283)
(-48.674144231954386, 0.0205404540417537)
(67.52943477714412, -0.0148067339465492)
(-7.725251836937707, -0.128374553525899)
(-17.22075527193077, 0.0579718023461539)
(86.38222203472871, -0.0115756804584678)
(32.956389039822476, 0.0303291711863103)
(-26.666054258812675, -0.0374745199939312)
(26.666054258812675, 0.0374745199939312)
(80.09812862894512, -0.012483713321779)
(108.37571965167469, 0.00922676625078197)
(-95.8081387868617, -0.0104369581345658)
(20.37130295928756, 0.0490296240140742)
(-83.2401924707234, -0.0120125604820527)
(10.904121659428899, -0.0913252028230577)
(83.2401924707234, 0.0120125604820527)
(-202.62779103941682, -0.00493509709208483)
(89.52422093041719, 0.0111694646341736)
(29.81159879089296, -0.0335251350213988)
(58.10225475449559, 0.0172084874716279)
(-54.959678287888934, 0.0181921463218031)
(-64.38711959055742, -0.0155291838074613)
(-39.24443236116419, -0.0254730530928808)
(-14.066193912831473, -0.0709134594504622)
(-70.6716857116195, -0.0141485220648664)
(-73.81388060068065, 0.01354634434514)
(73.81388060068065, -0.01354634434514)
(-36.10062224437561, 0.0276897323011492)
(-58.10225475449559, -0.0172084874716279)
(42.38791356813192, -0.0235850682290164)
(-51.81698248727967, -0.019295099487588)
(-23.519452498689006, 0.0424796169776126)
(51.81698248727967, 0.019295099487588)
(-80.09812862894512, 0.012483713321779)
Intervalos de crecimiento y decrecimiento de la función:Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=−32.9563890398225x2=−20.3713029592876x3=23.519452498689x4=61.2447302603744x5=−45.5311340139913x6=48.6741442319544x7=36.1006222443756x8=−76.9560263103312x9=98.9500628243319x10=−89.5242209304172x11=17.2207552719308x12=92.6661922776228x13=4.49340945790906x14=54.9596782878889x15=67.5294347771441x16=−7.72525183693771x17=86.3822220347287x18=−26.6660542588127x19=80.0981286289451x20=−95.8081387868617x21=−83.2401924707234x22=10.9041216594289x23=−202.627791039417x24=29.811598790893x25=−64.3871195905574x26=−39.2444323611642x27=−14.0661939128315x28=−70.6716857116195x29=73.8138806006806x30=−58.1022547544956x31=42.3879135681319x32=−51.8169824872797Puntos máximos de la función:
x32=−86.3822220347287x32=−4.49340945790906x32=−42.3879135681319x32=−67.5294347771441x32=7.72525183693771x32=−92.6661922776228x32=14.0661939128315x32=70.6716857116195x32=64.3871195905574x32=95.8081387868617x32=−29.811598790893x32=−10.9041216594289x32=76.9560263103312x32=45.5311340139913x32=39.2444323611642x32=−98.9500628243319x32=−61.2447302603744x32=−394.267341680887x32=−48.6741442319544x32=−17.2207552719308x32=32.9563890398225x32=26.6660542588127x32=108.375719651675x32=20.3713029592876x32=83.2401924707234x32=89.5242209304172x32=58.1022547544956x32=−54.9596782878889x32=−73.8138806006806x32=−36.1006222443756x32=−23.519452498689x32=51.8169824872797x32=−80.0981286289451Decrece en los intervalos
[98.9500628243319,∞)Crece en los intervalos
(−∞,−202.627791039417]